Homology, Homotopy and Applications

Volume 13 (2011)

Number 1

Colocalization functors in derived categories and torsion theories

Pages: 75 – 88

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n1.a4

Author

Shoham Shamir (Department of Mathematics, University of Bergen, Norway)

Abstract

Let $R$ be a ring and let $\mathcal{A}$ be a hereditary torsion class of $R$-modules. The inclusion of the localizing subcategory generated by $\mathcal{A}$ into the derived category of $R$ has a right adjoint, denoted CellA. Recently, Benson has shown how to compute $\operatorname{Cell}_{\mathcal{A}}R$ when $R$ is a group ring of a finite group over a prime field and $\mathcal{A}$ is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where $\mathcal{A}$ is any hereditary torsion class on $R$. It is shown that for every $R$-module $M$ there exists an injective $R$-module $E$ such that: